Does anybody know how this is figured out?

Question

anonymous · Answer

attachment

hartnn · Answer

did you find f1 (x), f2(x), f3 (x).... ? 
i got a repetitive pattern, see whther u also get it.

anonymous · Answer

@hartnn I think I got that too

anonymous · Answer

f1 1/1-x
f2 x-1/x
f3  x

hartnn · Answer

yes, i got same
and then f4 =f1, f5 = f2 , f6 = f3
correct ?

anonymous · Answer

yes that's right, but I am confused about the last part?

hartnn · Answer

f3 (x) = f6 (x) = f9 (x) =.... f {3n} (x)
f 2013 (x) = f3 (x)
right ?? 
because 2013 is divisible by 3

anonymous · Answer

oh right I see that

hartnn · Answer

then just plug in x=2013 to find 
f2013 (2013)

anonymous · Answer

so do I solve 2013 by 3 what you mean by plugging in x=2013

hartnn · Answer

what u got f2013 (x) as ?

anonymous · Answer

0?

hartnn · Answer

:O
i thought you understood this : f2013 (x) = f3 (x)

anonymous · Answer

uh...oh I was getting the remainder of 3/2013

hartnn · Answer

f2013 (x) = f3 (x) = x 
 f2013 (2013) = 2013 
by replacing x by 2013.

anonymous · Answer

so I try f1 and f2

hartnn · Answer

lol what ?? 
you wanted f2013 (2013)
you got it = 2013
didn't u get all steps on how ?

hartnn · Answer

should i write all steps again ?

anonymous · Answer

oh lol sorry I am confusing myself

anonymous · Answer

if you want that will be awesome, thanks

hartnn · Answer

$$ f_1 = 1/(1-x) \ 
f_2 = (x-1)/x \ 
f_3 =  x \ 
 f_4 =f_1, f_5 = f_2 , f_6 = f_3 \ 
f_3 (x) = f_6 (x) = f_9 (x) =.... f _{3n} (x) = x  \ 
\text {since, 2013 is divisible by 3} , \ 
f_{2013} (x) = x  \\text {just replace x by 2013 } \f_{2013} (2013) =2013 $$

anonymous · Answer

@hartnn thank you so much this is really helpful